Weighted surface algebras

JOURNAL OF ALGEBRA

卷:505

Article

Erdmann, Karin¹  Skowronski, Andrzej² 

[1] Univ Oxford, Math Inst, ROQ, Oxford OX2 6GG, England

[2] Nicolaus Copernicus Univ, Fac Math & Comp Sci, Chopina 12-18, PL-87100 Torun, Poland

关键词: Syzygy;    Periodic algebra;    Self-injective algebra;    Surface algebra;    Tame algebra;   

DOI  :  10.1016/j.jalgebra.2018.02.033

来源: Elsevier

【 摘 要 】

A finite-dimensional algebra A over an algebraically closed field K is called periodic if it is periodic under the action of the syzygy operator in the category of A-A-bimodules. The periodic algebras are self-injective and occurred naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period 4. Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are also symmetric tame periodic algebras of period 4. The main results of this paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type with 2-regular Gabriel quivers [36]. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free